Optimal stable control for nonlinear dynamical systems: an analytical dynamics based approach

This paper presents a simple methodology for obtaining the entire set of continuous controllers that cause a nonlinear dynamical system to exactly track a given trajectory. The trajectory is provided as a set of algebraic and/or differential equations that may or may not be explicitly dependent on time. Closed-form results are also provided for the real-time optimal control of such systems when the control cost to be minimized is any given weighted norm of the control, and the minimization is done not just of the integral of this norm over a span of time but also at each instant of time. The method provided is inspired by results from analytical dynamics and the close connection between nonlinear control and analytical dynamics is explored. The paper progressively moves from mechanical systems that are described by the second-order differential equations of Newton and/or Lagrange to the first-order equations of Poincaré, and then on to general first-order nonlinear dynamical systems. A numerical example illustrates the methodology.

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

The goal of this paper is to introduce a new neural network architecture called Sigmoid Diagonal Recurrent Neural Network (SDRNN) to be used in the adaptive control of nonlinear dynamical systems. This is done by adding a sigmoid weight victor in the hidden layer neurons to adapt of the shape of the sigmoid function making their outputs not restricted to the sigmoid function output. Also, we introduce a dynamic back propagation learning algorithm to train the new proposed network parameters. The simulation results showed that the (SDRNN) is more efficient and accurate than the DRNN in both the identification and adaptive control of nonlinear dynamical systems.

Stochastic optimal control for non-linear dynamical systems under noisy observations

Suboptimal Control of Nonlinear Dynamic System with Unlimited Transition Process Time

We describe a general method for adaptive model-based control of nonlinear dynamic systems using neural networks, fuzzy logic and fractal theory. The new neuro-fuzzy-fractal method combines soft computing techniques with the concept of the fractal dimension for the domain of nonlinear dynamic system control. The new method for adaptive model-based control has been implemented as a computer program to show that our neuro-fuzzy-fractal approach is a good alternative for controlling nonlinear dynamic systems. We illustrate our new methodology with the case of controlling aircraft dynamic systems. For this case, we use mathematical models for the simulation of aircraft dynamics during flight. The goal of constructing these models is to capture the dynamics of the aircraft, so as to have a way of controlling this dynamics to avoid dangerous behavior of the system.

This paper is concerned with neural-learning control of nonlinear dynamical systems. Two control schemes using neural-learning techniques are introduced. One is adaptive neural control and the other is predictive neural control. An application of neural-learning control to an industrial combustion system is also included. (7 pages)

In this paper, the use of radial basis function network (RBFN) for simultaneous online identification and indirect adaptive control of nonlinear dynamical systems is demonstrated. The motivation of using RBFN comes from the simplicity of its structure and simpler mathematical formulation, which gives it an advantage over multi-layer feed-forward neural network (MLFFNN). Since most processes are nonlinear, the use of conventional proportional-integral-derivative controller is not useful. Most of the time plant’s dynamics information is not available. This creates another limitation on the use of conventional control techniques, which works only if plant’s dynamics information is available. The proposed controller is tested for parameter variations and disturbance effects. Simulation results showed that RBFN is able to capture the unknown dynamics as well as simultaneously able to adaptively control the plant. It is also found to compensate the effects of parameter variations and disturbances. The comparative analysis is also done with MLFFNN in each simulation example, and it is found that performance of RBFN is better than that of MLFFNN.

A new recurrent neural-network predictive feedback control structure for a class of uncertain nonlinear dynamic time-delay systems in canonical form is developed and analyzed. The dynamic system has constant input and feedback time delays due to a communications channel. The proposed control structure consists of a linearized subsystem local to the controlled plant and a remote predictive controller located at the master command station. In the local linearized subsystem, a recurrent neural network with on-line weight tuning algorithm is employed to approximate the dynamics of the time-delay-free nonlinear plant. No linearity in the unknown parameters is required. No preliminary off-line weight learning is needed. The remote controller is a modified Smith predictor that provides prediction and maintains the desired tracking performance; an extra robustifying term is needed to guarantee stability. Rigorous stability proofs are given using Lyapunov analysis. The result is an adaptive neural net compensation scheme for unknown nonlinear systems with time delays. A simulation example is provided to demonstrate the effectiveness of the proposed control strategy.

This paper deals with the precision control of tumbling multibody systems with uncertainties present in the descriptions of their mathematical models. A generic tumbling multibody system consisting of a rigid body with internal degrees of freedom is used. A two-step control methodology is developed. First, a nominal system is conceived that best approximates the actual physical system. An analytical dynamics-based control methodology is used to obtain the nominal control force that ensures that this nominal system satisfies the control requirements. This is done using the control methodology proposed by Udwadia (“Optimal Tracking Control of Nonlinear Dynamical Systems,” Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, Vol. 464, 2008, pp. 2341–2363). Second, an additional compensating generalized control force is designed to ensure that the actual controlled (uncertain) system tracks the trajectories of the nominal system so that the control requirements are also met by the actual system. This paper deals primarily with the second step and its combination with the first. Uncertainties in both the description of the system as well as the forces acting on it are considered. No linearizations or approximations are made in either of the steps, and the full nonlinear dynamical system is considered. The efficacy of the control methodology is demonstrated by applying it to two tumbling uncertain multibody dynamical systems.

<p><strong>Context and relevance</strong>. The tasks of synthesizing optimal control for nonlinear dynamic systems with constraints and uncertainties remain computationally challenging, especially in aerospace applications. Reinforcement learning is considered a practical tool for building feedback and/or accelerating planning when classical methods are difficult to apply. <strong>Objective</strong>. To systematize classes of algorithms for optimal control tasks and identify criteria for selecting an approach for a specific problem. <strong>Hypothesis</strong>. Practical applicability is ensured by correct formulation and consideration of requirements for control continuity, data, safety, and robustness; combined solutions are most effective. <strong>Methods and materials</strong>. A review and comparative analysis of families of different reinforcement learning algorithms was performed. <strong>Results</strong>. Actor-critic remains the basis for continuous control, while alternatives increase selective efficiency but are sensitive to model and data coverage errors. <strong>Conclusions</strong>. The most promising are hybrid architectures that combine reinforcement learning with basic controllers and ensure controlled compliance with constraints. The choice of method should be determined not only by quality, but also by safety, robustness, and computational cost.</p>

Optimal control of nonlinear dynamical systems with fuzzy uncertainties

This paper analyzes, theoretically and by computer simulation, the capabilities and limitations of neural networks to be used for the identification and optimal control of nonlinear dynamical systems. It is shown how neural networks can be efficiently trained to identify the forward and inverse dynamics of systems and also to work as optimal controllers which minimize some peformance measures of the system to be controlled. The performance of neural networks when applied to the optimal control of nonlinear vehicle suspensions is analyzed and compared with the performance of passive suspensions and that of active suspensions with linear (LQ) controllers designed by linearizing the nonlinear characteristics of the suspensions around the equilibrium point. It is found that nonlinear vehicle suspensions with neuro-control show better performance than suspensions controlled with conventional LQ regulators. Some issues about the implementation of neural networks to improve its convergence and generalization properties are analyzed.

This paper is concerned with tele‐presence and tele‐operation of autonomous vehicles over the Additive White Gaussian Noise (AWGN) channel. This wireless communication channel is subject to transmission noise and transmission power constraint. An encoder, decoder and controller are proposed by implementing a novel linearization method for linearizing the nonlinear dynamic systems at operating points. For the linearized systems, the encoder, decoder and controller are implemented that were previously proposed for output tracking as well as stability of linear dynamic systems. Two novel linearization methods are proposed and their satisfactory performances are proved for output tracking as well as stability of nonlinear dynamic system. The first method is based on the fixed linearization rate and the second one is based on the variable (optimal) linearization rate. The decoder that is based on the second method is in fact the extended Kalman filter with the optimal linearization rate. The performances of these two methods are compared with each other and the other proposed methods by implementing them to the problems of the tele‐presence and tele‐operation of autonomous vehicles over the AWGN channel. Their satisfactory performances are illustrated. It is illustrated that the second method has higher computational complexity with slightly better performance and is applicable to a larger class of reference trajectories.

In this paper an automatic learning based adaptive approach elaborated for the control of nonlinear dynamic systems is analyzed from various points of view concerning the imprecision and incompleteness of the available system model used by the controller. The proposed approach is compared to the most elaborated classical techniques using Lyapunov functions and dynamic models that are exact in their mathematical form but imprecise in their parameters, yield globally and asymptotically stable solutions but do not allow the presence of permanent external perturbations. It is shown that the novel control allows both numerical imprecision and enduring external disturbances unknown by the controller, but generally cannot guarantee global stability. It is also shown that its simple structure makes it a prospective candidate for the control of fractional order dynamic phenomena in which the conventional techniques based on the application of integer order time-derivatives of quadratic Lyapunov functions have great difficulties. The possible implementation of the proposed method is mathematically tackled and expounded from various backgrounds as the application of Cauchy Sequences in complete, linear, normed metric (Banach) spaces, and the use of coupled differential equations that may also be obtained from simple quadratic Lyapunov functions the decreasing nature of which generally can be guaranteed only within bounded regions. The operation of the proposed method is illustrated by simulation examples made for integer and fractional order dynamical systems as examples.KeywordsLyapunov FunctionAdaptive ControlFractional OrderFractional DerivativeInput Multiple OutputThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.